An abstract system of congruence (ASC) is simply an equivalence relation on the power set of a finite set F satisfying some nondegeneracy conditions. Given such an ASC and an action of a group G on a set X, a realization of the ASC is a partition of X into pieces indexed by F such that whenever two subsets A, B are asc-equivalent, the corresponding subsets X_A and X_B of X can be translated to one another in the action. Familiar notions like paradoxical decompositions can be easily formalized and refined by the ASC language. Wagon, upon isolating this notion, characterized those ASCs which can be realized by rotations of the sphere. He asks whether there is an analogous characterization for realizing ASCs using partitions with the property of Baire. We provide such a characterization. This is joint work with Andrew Marks and Spencer Unger.